Prediction Model

We use the national survey data to fit a generalized mixed effects model:

\[\widehat{y}=Pr(y_i=1) = logit^{-1}(\alpha_{j[i]} + \beta x_i + \epsilon_{i})\]

where

\[ logit^{-1}(\alpha) = \frac{exp(\alpha)}{exp(\alpha) + 1}\]

where distributions drawn with mean zero and estimated variance (below are the individual-level predictive variables - random effects variables):

\[ \begin{aligned} \alpha ^{race}_{j} \sim N(0, \sigma ^{2}_{race}), for j= 1,...,3 \\ \alpha ^{age}_{k} \sim N(0, \sigma ^{2}_{age}), for k= 1,...,5 \\ \alpha ^{education}_{l} \sim N(0, \sigma ^{2}_{education}), for l= 1,...,4 \\ \alpha ^{gender}_{m} \sim N(0, \sigma ^{2}_{gender}), for m= 1,...,2 \\ \alpha ^{own_rent}_{or} \sim N(0, \sigma ^{2}_{own_rent}), for or= 1,...,2 \\ \alpha ^{state}_{d} \sim N(0, \sigma ^{2}_{state}), for d= 1,...,50 \\ \alpha ^{Hispanic}_{h} \sim N(0, \sigma ^{2}_{Hispanic}), for h= 1,...,2 \\ \alpha ^{hhincome}_{hhi} \sim N(0, \sigma ^{2}_{hhincome}), for hhi= 1,...,5 \\ \end{aligned} \]

In the model, an individual’s response to a particular survey question (the outcome variable) is a function of their individual level demographic variables (random effects with varying intercepts), state grouping variables (non random effects), and interactions. In the case of gas use in the home and other particular questions in the pr1928 survey, traditional predictive variables like political party identification and educational attainment are much less predictive than say region/state, percent that live in urban/suburban, owner/renter, etc. As a result, the statistical models also include housing characteristics to serve as state-level grouping variables such as heating fuel (percent of homes that have utility gas hookups in a state), urban/rural percentages, and housing types. For state-grouping level covariates we also included 2016 election returns to inform the statistical models, which is a strong predictor of state aggregated public opinion.

Model Specs:

For individual \(i\), with the following indexes for each demographic variable in the model, the model can be defined:

\[\begin{aligned} \widehat{y}=Pr(y_i=1) = logit^{-1}(\beta_0 + \alpha^{race}_{j[i]} + \alpha^{age}_{k[i]} + \alpha^{education}_{l[i]} + \alpha^{state}_{d[i]} + + \alpha^{ownerRenter}_{or[i]} \alpha^{gender}_{m[i]} + \\ \alpha^{Hispanic}_{h[i]} + \alpha^{hhincome}_{hhi[i]} + \alpha election_{2016} + \alpha CensusHouseHeatingFuel{5yearestimate} + \alpha PercentStateUrban \end{aligned} \]

The prediction for each strata, that is, the prediction for each “type of person” is weighted by the population frequency of the cell.

Sub-national cross tabs

To develop cross tabs for particular a demographic group, we first calculate the proportion of that demographic type in a given geographic region, we then filter for the specific subgroup (different levels within categories such as age, race, etc.) and develop new post-stratification weights by taking the cell weighted proportion and dividing it by the sum of the weights of that subgroup. In the final step, we sum the post-stratified predicted probabilities for each subgroup. If a demographic type does not have valid Census proportions (such as proportions of self identified Republicans in a given state, which isn’t tracked by the Census) we first develop an MRsP model to calculate the proportion of individuals that fit that demographic type, much like predicting any other survey question or outcome variable. We then use those proportions as given population percentages in a given state and can then filter, divide the sum of the weights of that variable, post-stratify, and sum across each cell.

We follow the process below to estimate opinions of a given subgroup.

\[m = poststratification weights of sub-population k\] \[ \begin{aligned} \widehat{y}=Pr(y_i=1) = logit^{-1}(\alpha_{j[i]} + \beta x_i + \epsilon_{i}) \\ logit^{-1} = \frac{exp(a)}{1 + exp(a)} \\ \widehat{weightedStatePred}= \sum_{k=1}^{n} logit^{-1}(\alpha_{j[i]} + \beta x_i + \epsilon_{i}) * \frac{m_i}{\sum m} \end{aligned} \]

Post-stratification & Confidence Intervals

To develop final estimates and error associated with the statistical models, we utilize statistical bootstrapping. Bootstrapping allows us to check the stability of the results.

To estimate the overall margin of error for each state, we run bootstrap samples for each answer option in the survey and calculate the confidence interval at the 95% confidence level. We then take the mean of all confidence intervals for a given state as the final margin of error.

Because each estimate (for a specific answer choice in the survey) is a modeled outcome variable with its own corresponding confidence interval, questions may or may not add up to 100%. If they do not, we normalize results to help with interpretation.